Figure 2 shows a flag XYWZ - Edexcel - A-Level Maths Pure - Question 7 - 2018 - Paper 4

The area of the flag consists of the area of triangle XYZ and the area of the sector ZYW.

First, we calculate the area of triangle XYZ using the formula: $A_{triangle} = \frac{1}{2} \times b \times h$

In this case:

• Base ($b$) = $XY = 7 \text{ cm}$
• Height ($h$) = YZ = $5 \text{ cm} \times \sin(0.7)$ (approximating using the sine of the angle)

Calculating: $h = 5 \times \sin(0.7) \approx 5 \times 0.644 = 3.22 \text{ cm}$

Now, substituting back: $A_{triangle} = \frac{1}{2} \times 7 \times 3.22 \approx 11.27 \text{ cm}^2$

Now summing the areas: $\text{Total area} = A_{triangle} + A_{sector} = 11.27 + 8.75 = 20.02 \text{ cm}^2$

Thus, the area of the flag is 20.02 cm².

To calculate the perimeter of the flag XYWZ, we will sum the lengths of each side:

• XY = 7 cm
• YW = 5 cm
• The length of YZ can be found using the Pythagorean theorem in triangle XYZ:

1. Find length YZ: Using $YZ = 5 \sin(0.7) \approx 3.22 \text{ cm}$ as calculated previously.
2. Find length ZX: Using $ZX = \sqrt{ZY^2 + XY^2} = \sqrt{(5)^2 + (7)^2} = \sqrt{25 + 49} = \sqrt{74} \approx 8.60 ext{ cm}$.

Therefore, the total perimeter is: $\text{Perimeter} = XY + YW + YZ + ZX = 7 + 5 + 3.22 + 8.60 = 23.82 \text{ cm}$

Thus, the length of the perimeter XYWZ is approximately 23.82 cm.